Question

In how many ways can the letters of the word vowels be arranged, if the letters o and e can occupy odd places.

Answer 1

Hint: This is a question based on the arrangement of alphabets. In this question we have to arrange the letters of the given word satisfying the condition that the vowels of the given word occupy odd places. So the first step is to count and write the total number of places and then mark odd and even places. After this select two odd places out of three and then arrange the two vowels. Finally arrange the remaining alphabet i.e. consonant.

Complete Step-by-Step solution:
In the question, it is given that we have to arrange the letters of word vowels such that the letters o and e occupy odd places.
So, we will first write the total number of places and mark the odd and even places.

   

The odd places are 1,3 and 5. We have to arrange the vowels o and e at these places.
For odd places total arrangements = selection of 2 places out of 3 and then arranging the two vowels at these places $ = {}^3{{\text{C}}_2} \times 2$ǃ
For the rest of places the arrangement is 4!
   Total ways of arrangement at rest places $ = 4 \times 3 \times 2 \times 1$
$\therefore$ Total number of arrangement of word vowels such that letters o and e occupy odd places $ = { }^3{{\text{C}}_2} \times 2! \times 4!$ = $\dfrac{3}{2} \times \dfrac{2}{1} \times 2 \times 4 \times 3 \times 2 \times 1$ = $144$

Note: Before solving this type of question, you should know how to arrange letters such that they do not repeat itself. In this question, letters of the vowels are to be arranged given a condition that o and e occupy odd places. So first of all make a rough diagram indicating the total number of places and mark odd and even places. Now we have two vowels o and e but three odd places, so first select 2 places out of 3 places and then arrange the two vowels. Finally arrange the remaining letters at remaining place